Gears Force and Power
From CRCycling
This article is the first in a series about the physics of getting around on a racing bicycle. This article reviews the basics of Force, Power, and Gearing.
Contents |
Bicycle Gearing
One of the earliest bicycle designs (developed in the 1860s), the “ordinary” or the “penny farthing”, used a direct drive mechanism. The cranks were directly attached to the drive wheel. One turn of the cranks causes one revolution of the wheel. Since the wheel circumference is much larger than the pedal circumference, the wheel moves a greater distance than the pedals in one revolution. Therefore, the mechanism multiplies the velocity of the crank.
In one revolution of a 175 mm crank, the pedals travel 2 x pi x 175 mm = 1099 mm, while a point on the wheel travels 2 x pi x 700 mm = 4398 mm. The velocity of the cranks is multiplied by the ratio of the wheel radius by the crank radius. In this case 700 mm / 175 mm = 4. The force exerted by the rider, however, is decreased by the same ratio. To deliver 10 pounds of force to the ground, a rider must exert 40 pounds of force on the pedals.
This force and velocity relationship can be understood by comparing it to the simple lever shown in Figure 1 to the right. When the fulcrum is placed near the object to be moved, force is multiplied at the expense of velocity. The lever on the right side of the fulcrum is 3 times the length of on the object side, so the force is multiplied by 3. When the fulcrum is far from the object to be moved, velocity is multiplied at the expense of force.
The ordinary bicycle design had obvious flaws. The size of the front wheel is limited by a rider’s leg length. Also, the high center of gravity is inherently dangerous. A rider could easily be launched over the handlebars by a bump in the road. What we call a face plant, they used to call "coming a cropper".
The modern “safety bicycle” replaced the huge front wheel of the ordinary with a chain and gear system. The gears are a mechanism for virtually changing the size of the rear wheel. The gears multiply the velocity of the crank the same way the huge wheel of the ordinary multiplies the crank velocity. If the rear wheel radius is 335 mm (for a 700C wheel) and the gear ratio is 42/20 = 2.1, the crank velocity is multiplied by:
- 335 x 2.1 / 175 = 4
So, with this gear combination a point on the familiar 700C rear wheel covers the same distance as a point on a 54" Ordinary wheel in one crank revolution.
The Ordinary force relationship holds true for the gear system of the safety bicycle. When the gear ratio is greater than 1, the force that's delivered at the pedals is diminished. To exert 10 pounds of force on the ground while pedaling a 42x20 gear, it's necessary to exert 40 pounds at the pedal.
Energy, Force, Power
A few simple concepts from classical mechanics, help shed some light on the day-to-day experiences of cycling. Cycling is all about the efficient conversion of chemical energy stored in the human body into the energy of motion. This can be described with a handful of equations from classical mechanics.
Isaac Newton’s famous equation states that Force is proportional to mass and acceleration, i.e.:
- Force = mass x acceleration
The SI unit for Force is the Newton. 1 Newton = 1 kilogram-meter/second2.
In classical mechanics, Energy is force applied through a distance. The SI unit for Energy is the Joule. (Food energy is commonly measured in kilo-calories. 1 kCalorie = 4,184 Joules.)
Power is the rate at which Energy is stored or expended; it is Energy per unit time. The SI unit for Power is Watts. 1 Watt = 1 Joule/second. (1 Horsepower = 745 Watts.)
Some other useful relationships are:
- Power = Force x Velocity
so...
- Force = Power / Velocity
Gravitational potential energy is force through a distance applied against the Earth's gravity. The gravitational potential energy of an object increases as it's moved away from the center of the Earth--as you climb a hill on a bike, for example.
- Ep = mass x gravitational acceleration * elevation
Kinetic energy is the energy of motion. The kinetic energy of a moving object is:
- Ek = 1/2 mass x velocity2
When you descend a hill, gravitational potential energy is converted into kinetic energy.
Examples
Near the surface of the Earth, all of the atoms in the planet (all 5.9742 × 1024 kilograms of them) gravitationally attract objects so an unsupported object accelerates toward the center of the planet at 9.8 meters/second per second. That is, if you drop a baseball from a balcony, 1 second later it will be moving toward the ground at 9.8 meters/second. In two seconds it will be 19.6 m/s.
If you lift a 10 kilogram weight 1 meter, you have imparted 10 kilograms * 9.8 m/s2 * 1 meter = 98 Joules of Energy into the dumbbell. If the move is completed in 1 second, the average Power expended during the move was 98 Joules / 1 second = 98 Watts. The average force is 98 Newtons.
These simple relationships are handy for calculating the power and force exerted by a rider to propel a bike.
A rider + bike that weighs in at 100 kg (that’s about 220 pounds) that climbs Kirtland Chardon Road and gains 475 feet (144.78 meters) in elevation has expended:
- 100 kg x 9.8 x 144.78 = 141,884 Joules (about 34 kCalories)
If the trip took about 8 minutes, the average power is approximately:
- 141,884 / (8 * 60) = 295 Watts
Note, this is only a meaningful approximation at low speeds (below 12 mph) where wind resistance is negligible, and also ignores rolling resistance, and friction losses in the drivetrain. Rolling resistance and friction losses are typically pretty small (on smooth pavement) compared to the other forces opposing a rider's motion.
If the rider made the 1.3 mile climb in 8 minutes, the rider travelled at 9.75 mph, or 4.358 meters/second. Since Power = Force x Velocity, we can easily compute the average Force exerted at the wheel. It’s 67.68 Newtons.
As discussed in the previous section, the velocity of the wheel is magnified by the gear system, and the force transmitted to the wheel is diminished. The reverse is also true; force applied at the wheel is magnified at the pedal. If the rider has a 175 mm crank, and rides a 39x27, and the radius of the rear wheel of the bike is 335 mm, to create 67.68 Newtons at the wheel, the rider will have to apply:
- (335 mm * (39 / 27)) / 175 = 2.765 x 67.68 N = 187 Newtons
The average force exerted by the rider at the pedal is 187 Newtons. That’s equivalent to lifting a 42 pound weight with your legs every instant during the climb.
Next
These simple equations provide an insight into the average behavior of the bicycle in certain circumstances (climbing a hill slowly), however, they don't provide a detailed description of the bike and rider at any particular moment. Also, they fail to provide useful information about other circumstances, like riding at high speed on level ground. The next article in the series will cover the mechanics of pedaling and rider position on the bike.
Further Reading
- History of the Bicycle
- Wilson, David Gordon. Bicycling Science
- Excel Spreadsheet with Crank Force Calculation
